Irreducible representations of Lie algebra extensions

Richard E. Block
1974 Bulletin of the American Mathematical Society  
This note announces three density theorems involving representations of Lie algebras and associative algebras. The first theorem describes the irreducible (possibly infinite dimensional) representations p of a Lie algebra g with an ideal ï such that the restriction of p to ï has some absolutely irreducible quotient representation. The second result is an embedding theorem for the irreducible representations of the Weyl algebras A niC over C (A n>c^C [t l9 • • • , t n , d/d^, • • • , 9/9^J, the
more » ... • • • , 9/9^J, the associative algebra of partial differential operators on n variables with coefficients in the polynomial ring C[t l9 • • • , t n ]). Our result is a sort of algebraic analogue of the uniqueness of the Heisenberg commutation relations, and has an application to irreducible representations of nilpotent Lie algebras via Dixmier's theory [5] . The third theorem describes the differentiably simple algebras having a maximal ideal. This result unifies the author's theorem [3] on differentiably simple rings with a minimal ideal, and Guillemin's theorem [7], [2] on the structure of a nonabelian minimal closed ideal of a linearly compact Lie algebra.
doi:10.1090/s0002-9904-1974-13547-0 fatcat:25rju7d5jbb6dojeacslwamdwe